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Should Scoring Rules be "Effective"?

  • Robert F. Nau

    (A. B. Freeman School of Business, Tulane University, New Orleans, Louisiana 70118)

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    A scoring rule is a reward function for eliciting or evaluating forecasts expressed as discrete or continuous probability distributions. A rule is strictly proper if it encourages the forecaster to state his true subjective probabilities, and effective if it is associated with a metric on the set of probability distributions. Recently, the property of effectiveness (which is stronger than strict properness) has been proposed as a desideratum for scoring rules for continuous forecasts, for reasons of "monotonicity" in keeping the forecaster close to his true probabilities, since in practice the forecast must be chosen from a low-dimensional set of "admissible" distributions. It is shown in this paper that what effectiveness implies, beyond strict properness, is not a monotonicity property but a transitivity property, which is difficult to justify behaviorally. The logarithmic scoring rule is shown to violate the transitivity property, and hence is not effective. The L 1 and L \infty metrics are shown to allow no effective scoring rules. Some potential difficulties in interpreting admissible forecasts are also discussed.

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    File URL: http://dx.doi.org/10.1287/mnsc.31.5.527
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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 31 (1985)
    Issue (Month): 5 (May)
    Pages: 527-535

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    Handle: RePEc:inm:ormnsc:v:31:y:1985:i:5:p:527-535
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